Quantum information geometric foundations: beyond the spectral paradigm

Abstract

In the last decade there were proposed several new information theoretic frameworks (in particular, symmetric monoidal categories and "operational" convex sets), allowing for an axiomatic derivation of finite dimensional quantum mechanics as a specific case of a larger universe of information processing theories. Parallel to this, there was an influential development of quantum versions of bayesianism and causality, and relationships between quantum information and space-time structure. In the face of structural problems encountered when moving beyond finite dimensional quantum mechanics, as well as the lack of a mathematically and predictively sound nonperturbative framework for quantum field theories, a question appears: which of the existing structural assumptions of quantum information theory should be relaxed, and how? In this talk I will present a new approach to the information theoretic foundations of a "general" quantum theory (i.e., beyond quantum mechanics), that is a specific answer to the above question, with a hope to reconstruct both emergent space-times and emergent QFTs. Its mathematical setting is based on using quantum information geometry and integration over noncomutative algebras as structural and conceptual replacements of spectral theory and probability theory, respectively. This corresponds to a paradigmatic change: considering expectation values as more fundamental than eigenvalues. We construct a nonlinear generalisation of quantum kinematics using quantum relative entropies and spaces of states over W*-algebras. Unitary evolution is generalised to nonlinear hamiltonian flows, while Bayes' and Lueders' rules are generalised to constrained relative entropy maximisations. Combined together, they provide a framework for nonlinear causal inference (information dynamics), that is a generalisation and replacement of completely positive maps. As a result, we construct a large class of information processing theories, containing Hilbert space based QM and probability theory as two special cases. On the conceptual level, we propose a new approach to quantum bayesianism, that is ontically agnostic, intersubjective, and concerned with the relationships between experimental design, model construction, and their mutual predictive verifiability. Finally, we propose a procedure for the emergence of space-times from the geometry of quantum correlations and quantum causality structure, and discuss (briefly) the possibility of reconstructing emergent QFTs.